Magnetic Anisotropy of Polycrystalline High-Temperature Ferromagnetic MARK Mnxsi1-X (X ≈0.5) Alloy ﬁlms ⁎ A.B

Journal of Magnetism and Magnetic Materials 429 (2017) 305–313

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Journal of Magnetism and Magnetic Materials

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Magnetic anisotropy of polycrystalline high-temperature ferromagnetic MARK MnxSi1-x (x ≈0.5) alloy ﬁlms ⁎ A.B. Drovosekova, , N.M. Kreinesa, A.O. Savitskya, S.V. Kapelnitskyb,c, V.V. Rylkovb,d, V.V. Tugushevb,e, G.V. Prutskovb, O.A. Novodvorskiif, A.V. Shorokhovaf, Y. Wangg, S. Zhoug a P.L.Kapitza Institute for Physical Problems RAS, Kosygina St. 2, 119334 Moscow, Russia b National Research Centre “Kurchatov Institute”, Kurchatov Sq. 1, 123182 Moscow, Russia c Institute of Physics and Technology RAS, Nakhimovsky av. 36, build 1, 117218 Moscow, Russia d Kotel'nikov Institute of Radio Engineering and Electronics RAS, 141190 Fryazino, Moscow Region, Russia e Prokhorov General Physics Institute RAS, Vavilov St. 38, 119991 Moscow, Russia f Institute on Laser and Information Technologies RAS, Svyatoozerskaya St. 1, 140700 Shatura, Moscow Region, Russia g Helmholtz-Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics and Materials Research, Bautzner Landstrasse 400, 01328 Dresden, Germany

ARTICLE INFO ABSTRACT

Keywords: A set of thin film MnxSi1-x alloy samples with different manganese concentration x ≈0.44−0.63 grown by the fi Magnetic MnxSi1-x alloy lms pulsed laser deposition (PLD) method onto the Al2O3 (0001) substrate was investigated in the temperature Magnetic anisotropy range 4–300 K using ferromagnetic resonance (FMR) measurements in the wide range of frequencies Ferromagnetic resonance ( f =7−60GHz) and magnetic fields (H =0−30kOe). For samples with x ≈0.52−0.55, FMR data show

clear evidence of ferromagnetism with high Curie temperatures TC ∼ 300 K. These samples demonstrate a complex magnetic anisotropy described phenomenologically as a combination of the essential second order easy plane anisotropy contribution and the additional fourth order uniaxial anisotropy contribution with easy direction normal to the film plane. The observed anisotropy is attributed to a polycrystalline (mosaic) structure of the films caused by the film-substrate lattice mismatch. The existence of extra strains at the crystallite boundaries initiates a random distribution of local in-plane anisotropy axes in the samples. As a result, the symmetry of the net magnetic anisotropy is axial with the symmetry axis normal to the film plane. The principal features of the observed anisotropy are explained qualitatively within the proposed microscopic model.

1. Introduction materials in spintronic applications. At the same time, the fabrication of well-reproducible homogeneous magnetic alloys with high Mn Development of Si based magnetic semiconductor materials for content x ≈0.35 is diﬃcult because of the variety of stable phases of spintronic applications attracts a lot of attention, since these materials high MnSiy silicides (not less than ﬁve) with the close content of can be easily incorporated into the existing microelectronic technology components (y = 1.72 − 1.75) [6,7].

[1]. In particular, Si-Mn alloys demonstrating unusual magnetic and In contrast, nonstoichiometric MnxSi1-x alloys with high Mn transport properties [2–8] have especial interest to engineer non- content (x ≈0.5, i.e. close to stoichiometric MnSi) are not inclined to conventional integrated-circuit elements. a phase segregation and formation of isolated magnetic precipitates, so However, there exist signiﬁcant technological and fundamental they seem more promising for spintronic applications than dilute obstacles to adapt Si-Mn based elements to the needs of spintronic. MnxSi1-x alloys. Recently we have found that in thin ﬁlms of such

At relatively low Mn content in MnxSi1-x alloys (x = 0.05 − 0.1), the concentrated alloys, the Curie temperature TC increases by more than ferromagnetism (FM) at above room temperature has been revealed. an order of magnitude as compared with bulk MnSi (TC ≈30K) [6]. But these alloys turn out to be strongly inhomogeneous materials due Comparative studies of anomalous Hall effect and transverse Kerr to their phase segregation, leading to formation of isolated magnetic effect showed that the ferromagnetic transition in MnxSi1-x MnSi1.7 precipitate nanoparticles with the Mn content x ≈0.35 in Si (x ≈ 0.52 − 0.55) alloys occurring at T ∼ 300 K, has a global nature ff matrix [2]. In such alloys, anomalous Hall e ect testifying the spin and is not associated with the phase segregation [7]. Besides high TC polarization of carriers is absent, that makes impossible to use these values, the films investigated in [6,7] show large values of saturation

⁎ Corresponding author. E-mail address: [email protected] (A.B. Drovosekov). http://dx.doi.org/10.1016/j.jmmm.2017.01.022 Received 22 September 2015; Received in revised form 21 December 2016; Accepted 8 January 2017 Available online 10 January 2017 0304-8853/ © 2017 Elsevier B.V. All rights reserved. A.B. Drovosekov et al. Journal of Magnetism and Magnetic Materials 429 (2017) 305–313 magnetization reaching ≈400 emu/cm3 at low temperatures. The ﬁ observed magnetization value corresponds to ≈1.1μB /Mn, that signi - cantly exceeds the value 0.4μB /Mn typical for bulk MnSi crystal [9]. High temperature FM in MnxSi1-x (x ≈0.5) alloys has been qualitatively interpreted [6,7] in frame of the early proposed model for dilute MnxSi1-x alloys [3], i.e. in terms of complex defects with local magnetic moments embedded into the matrix of itinerant FM.

However, many details of FM order in MnxSi1-x(x ≈0.5) alloys are still not completely clear due to insufficient experimental studies. In particular, there are no data on their magnetic anisotropy features. In the present work, thin MnxSi1-x (x ≈ 0.52 − 0.55) films are investigated by the ferromagnetic resonance (FMR) method which is powerful for providing valuable information about magnetic anisotropy peculiarities of thin film magnetic materials, in particular, like dilute magnetic semiconductors (see [10] and references therein). Our studies are largely focused on the position and shape of FMR signal, while the analysis of the refinements of the line width data providing additional information on the magnetic inhomogeneity and relaxation rate of fi Fig. 1. Temperature dependence of the resonance eld HTres( ) at the frequency magnetization is not presented in this work. Besides, to study the 17.4 GHz for samples with different manganese concentration. The dashed line details of crystalline and magnetic microstructures of our films we corresponds to the calculated position of paramagnetic resonance for g-factor g=2.14. perform in this work the atomic force microscopy (AFM) and magnetic The inset demonstrates examples of the experimental resonance spectra at T=77 K. The force microscopy (MFM) investigations. We hope that AFM and MFM field is applied in the film plane. methods allow for additional understanding of the FMR results, since these methods are able to reveal the “local” effects of the crystal and temperature increases, the resonance field also increases and at magnetic texture of the film on the origin of magnetic anisotropy T ∼ 300 K reaches the value corresponding to the paramagnetic established in FMR measurements. resonance situation. The observed absorption lines have the Lorentz- like shape, and the resonance field anisotropy in the film plane is 2. Samples and experimental details absent. The shift of the FMR absorption line revealed in our measurements We studied six samples with manganese content in the range (shown in Fig. 1) clearly indicates the presence of FM moment in x ≈ 0.44 − 0.63. The 70 nm thick film samples were produced by the samples with x ≈ 0.52 − 0.55 at low temperatures, in agreement with Ref. [6]. When the external magnetic field H is applied in the plane of a pulse laser deposition (PLD) method on Al2O3 (0001) substrates at 340 °C. The composition of the films was testified by X-ray photoelec- thin FM film with magnetization M, the FMR frequency f may be tronic spectroscopy (for details see [6]). described by the phenomenological formula (see Appendix A): The structural properties of the samples were studied by X-ray f =(+)γHH KM. diffraction (XRD) analysis using a Rigaku SmartLab diffractometer (1) without monochromators and a diaphragm before the detector. In this Here K is the effective easy plane magnetic anisotropy coefficient; K M 9 case, intensity of the direct beam was as high as 1.5·10 pulses/s. is the effective field of easy plane magnetic anisotropy. This field Additionally, we performed room temperature AFM and MFM inves- describes the effect of two factors on the FMR spectra: 1) the shape tigations in the sample MnxSi1-x (x ≈0.52) having the most pro- magnetic anisotropy depending on the form of the sample and 2) the nounced high-TC FM, using microscope SmartSPM (AIST-NT). crystal structure driven magnetic anisotropy which is due to relativistic FMR spectra of the obtained samples were studied at temperatures interactions between electrons and ions in the sample material. T = 4 − 300 K in the wide range of frequencies ( f =7−60GHz) and Following a simplest model used in Ref. [12] it is easy to obtain: magnetic fields (H =0−30kOe). To detect the resonance absorption 2K signal, magnetic field dependences of the microwave power transmitted K MπM=4 + 1 , (2) through the cavity resonator with the sample inside were measured at M ff constant frequency. Measurements were carried out for di erent where K1 is the phenomenological constant of the second-order easy orientations of the magnetic field with respect to the film plane. plane crystal structure driven magnetic anisotropy. In the absence of

K1, Eq. (1) transforms into the well known Kittel formula for the 3. Experimental results and discussion applied field H lying in the film plane [13]. According to Eq. (1), a reduction of FM moment of the film with 3.1. FMR measurements increasing temperature leads to a decrease of the FMR line shift with respect to the paramagnetic resonance situation. Thus FMR data The dependence of the resonance field on temperature in case of the (Fig. 1) confirm the FM order up to T ∼ 300 K in samples with Mn field applied in the film plane is shown in Fig. 1 for several samples. concentration x ≈ 0.52 − 0.55. Films with Mn concentration x ≈0.44 and x ≈0.63demonstrate the Dependences of the FMR frequency on the magnetic field applied in resonance absorption peak in the field corresponding to a paramag- the film plane for samples with Mn concentrations 0.52 and 0.53 at netic resonance situation, i.e. f = γH, with the gyromagnetic ratio T=77 K are shown in Fig. 2.Atsufficiently high fields H > 5 kOe (i.e. in γ ≈ 3 GHz/kOe corresponding to the g-factor value g=2.14, which is in the high-frequency region f > 25 GHz), the f(H) dependence can be agreement with the value reported in Ref. [11] for the bulk MnSi single well approximated by Eq. (1) with the field-independent value crystal. The paramagnetism of the samples with x ≈0.44 and x ≈0.63 K M ≈ 8.7 kOe (at T=77 K both samples have about the same K M is observed in the temperature range 20 − 300 K, that is in accordance values). However, some deviations of the experimental points from the with the results of Ref. [6]. theoretical curve are observed at smaller fields. In the bottom inset of At low temperatures in the range of concentrations x ≈ 0.52 − 0.55, Fig. 2, the experimentally obtained field dependence of K M parameter the films show significant shift of absorption peak into the region of is shown. One can see that the value of K M parameter significantly smaller fields with respect to paramagnetic samples (Fig. 1). As the depends on the applied field at small fields H <5kOe, while it comes

306 A.B. Drovosekov et al. Journal of Magnetism and Magnetic Materials 429 (2017) 305–313

exposed in Appendix A, the large difference between K M and 4πM can be explained by the fact that the crystal structure driven second order easy plane magnetic anisotropy is comparable with the sample shape magnetic anisotropy. The similarity of the M(T) and K MT()curves means that K is almost temperature independent and consequently 2 K1()∼TMT (). The inset in Fig. 3 demonstrates the resulting temperature dependence of the K1 constant. To obtain further insight into the peculiarities of the magnetic anisotropy of our system, the f(H) dependences were investigated for samples with x ≈0.52 and x ≈0.53at T=4.2 K, when the applied field was perpendicular to the film plane (Fig. 4). Within the phenomenological model exposed in Appendix A, in this

case the FMR frequency f⊥ in the saturation regime has a linear dependence on the applied external ﬁeld:

f⊥ =(γH − KM⊥ ), (3) ff ffi where K⊥ is the e ective hard axis magnetic anisotropy coe cient, K⊥M is the effective field of hard axis magnetic anisotropy. Due to this anisotropy, the FMR line in perpendicular geometry is shifted to the region of larger fields comparing the paramagnetic case f = γH. This behavior is opposite to the case of the parallel geometry (Eq. (1)) where the FMR line was shifted to lower fields (Fig. 1). Eq. (3) is applicable in fi fi magnetic elds exceeding the saturation eld HS⊥= KM while below

this value f⊥ =0. Fig. 2. Dependence of the resonance frequency on the magnetic field applied in the film Without the crystal structure driven magnetic anisotropy contribu- plane for samples with x ≈0.52 and x ≈0.53at T=77 K. Points are experimental data; the tion, K⊥ ==4K π and Eq. (3) transforms into the Kittel formula for the line is the theoretical curve according to Eq. (1). The insets represent static magnetiza- FMR frequency, when applied field H lies normally to the thin film tion curves (the upper inset) and the value of the parameter K M in Eq. (1) as a function plane [13]. If we take into account only the second order crystal of magnetic field (the bottom inset). structure driven magnetic anisotropy contributions to describe the total ffi magnetic anisotropy in our system, the coe cient K⊥ coincides with K [12]. nigh unto a saturation at higher fields H >5kOe. In agreement with Eq. (3), the experimental f⊥ (H) dependences are The observed deviation of the K M parameter at low fields from the linear in the region of high frequencies and fields (Fig. 4). Nevertheless, constant value at high fields is probably due to a random distribution of ff the K⊥M parameter di ers from the K M parameter. It is seen from local magnetic anisotropy axes in the sample plane resulting in Fig. 4 that the experimental f⊥ (H) dependencies are poorly described fi inhomogeneity of magnetization at low applied elds. Indeed, static using for the K⊥M parameter the K M value obtained in the parallel magnetization curves (upper inset in Fig. 2) achieve a saturation in geometry. For both samples, the K⊥M parameter is less than K M but relatively high fields H ∼ 5 kOe. One can notice, however, that the K M exceeds the 4πM value obtained from static magnetization measure- parameter estimated from the FMR data increases as the magnetic field ments (see Table 1). decreases below 5 kOe, while the static magnetization diminishes. This Following the simplest phenomenological approach of Appendix A, ff ff contradiction clearly shows inapplicability of Eq. (1) in the region of the di erence between K⊥ and K can be attributed to the e ect of a small fields and indicates inhomogeneity of local magnetization and higher order uniaxial magnetic anisotropy of the sample. Here the term anisotropy. Below in this work, only FMR data obtained at high “uniaxial anisotropy” means that the magnetic energy expression has a frequencies f > 25 GHz are taken into account to estimate the magnetic uniaxial symmetry, i.e. it is invariant with respect to arbitrary rotations anisotropy parameters of the system. Such frequencies provide suffi- about the axis z normal to the sample plane. The anisotropy order is ciently high resonance fields H >5kOe, where the FMR data can be defined by the power of z-component of magnetization in the energy well approximated in frame of Kittel's formalism (Eq. (1)). expression. Temperature dependences of the K M parameter obtained by means Taking into account the second order easy plane anisotropy and of Eq. (1) for the films with Mn concentration x ≈ 0.52 − 0.55 are given neglecting higher than fourth order terms of uniaxial anisotropy (see ff fi in Fig. 3. For all the samples, the low temperature value of the K M Appendix A), the relation between the K⊥M and K M e ective elds has parameter is about 10 kOe (see Table 1). The K MT()curve for the the form: sample with x ≈0.52 has the Brillouin-like shape with T ≈ 300 K. Note C 4K that the Brillouin curve gives smaller T value than found from static K MKM=+2 , C ⊥ (4) magnetization measurements [6] (more precisely M(T) dependence can M be fitted within spin-fluctuation model [3]; see Fig. 3 and [6]). For the where K2 is the fourth-order constant of crystal structure driven fi lms with x ≥0.53, the curve K MT()is closer to the linear dependence: magnetic anisotropy. Thus, Eqs. (1)–(4) are similar to those used in this fact can be caused by a heterogeneity of samples that is also Ref. [14] with a little different definition of the K and K constants. fi 1 2 con rmed by a larger line width of the samples with x ≈ 0.53 − 0.55 in Note that a more complex description by means of two independent comparison with the case x ≈0.52 (see the inset in Fig. 1). fourth-order phenomenological constants is also possible by presuming Besides the experimental K MT()curves, Fig. 3 represents the a tetragonal-like character of the film distortion. However, it is beyond fi temperature dependences of the demagnetizing eld 4πM calculated the scope of the current paper. from static magnetization data at H=10 kOe applied in the film plane. The obtained low-temperature values of the K1 and K2 constants are The shapes of the K MT()and 4(πM T) curves are close to each other, given in Table 1. The positive sign of the K1 constant corresponds to a while the low temperature values of K M parameter exceed 4πM second-order easy plane contribution into the total magnetic aniso- considerably (about two times; see Table 1). These two values equalize tropy of the film. The negative sign of the K constant corresponds to a only in the vicinity of T . In frame of the phenomenological model 2 C fourth-order easy axis contribution into the total magnetic anisotropy

307 A.B. Drovosekov et al. Journal of Magnetism and Magnetic Materials 429 (2017) 305–313

Fig. 3. Temperature dependence of K M parameter (solid circles) obtained from FMR, and demagnetizing field 4πM (open circles) obtained from static magnetization data for samples with concentration x ≈ 0.52, 0.53 and 0.54. The dash dot line (blue) in the left plot represents Brillouin curve K MT( ) for spin 1/2; the solid line (red) is theoretical 4(πM T ) curve determined in [6] within the framework of the spin-fluctuation model [3]. Crosses in the right plot represent K MT( ) dependence for x ≈0.55. The insets show experimental temperature dependencies of the anisontropy constant K1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 1 Effective demagnetization, anisotropy fields and corresponding anisotropy constants deduced for samples with Mn content x ≈0.52, 0.53 and 0.54 at T=4.2 K using Eqs. (1– 4).

x 4πM K M K⊥M K1 K2

(kOe) (106 erg/cm3) 0.52 5.5 9.0 6.4 0.8 −0.3 0.53 4.1 10.8 8.7 1.1 −0.2 0.54 3.9 9.5 no data 0.9 no data

Fig. 5. Hysteresis loops of the sample with x=0.52 in case of the ﬁeld applied in the ﬁlm plane and normal to it at T=10 K.

of the film. As an additional demonstration of the role of magnetic anisotropy effects in our system, the low-temperature magnetization curve for the sample with x ≈0.52 was measured in the field applied perpendicular to the sample plane (Fig. 5). If only the second-order easy plane anisotropy takes place, the magnetization must depend on the mag- fi fi netic eld linearly below the saturation eld HS = KM. The anisotropy of higher orders results in a nonlinearity of the magnetization curve. In fi fi this case the saturation eld is de ned by HS⊥= KM. The experimental magnetization curve (Fig. 5) shows a smooth approach to saturation. Moreover, there is noticeable hysteresis of the fi M(H) curve (the coercive eld HC ≈0.65kOe). Linear extrapolation of the initial part of the magnetization curve leads to saturation field

HS ≈7.5kOe. The obtained value is smaller than the K M observed in Fig. 4. Dependence of the resonance frequency on the magnetic field applied normal to FMR experiments, but larger than the K⊥M. Thus, there is only some the film plane for samples with x=0.52 and x=0.53 at T=4.2 K. Points are experimental qualitative agreement of static magnetization and resonance data. The data; solid lines are the theoretical curves according to Eq. (3). The dashed (red) and static magnetization curves confirm the existence of significant second- dash-dotted (blue) lines would correspond to the samples with x=0.52 and x=0.53 order easy plane anisotropy, but allow for neither confirmation nor respectively if we considered K⊥MKM= (see details in text). The inset demonstrates rejection our assumption about the effect of the higher order contribu- examples of the experimental resonance spectra at frequency 49 GHz. (For interpretation tions of crystal structure driven magnetic anisotropy on the total of the references to color in this figure legend, the reader is referred to the web version of magnetic anisotropy of the system. this article.)

308 A.B. Drovosekov et al. Journal of Magnetism and Magnetic Materials 429 (2017) 305–313

Fig. 6. The X-ray diﬀraction pattern for the MnxSi1-x/Al2O3 structure with x ≈0.52. The inset shows the XRD curve for sample with x ≈0.63.

3.2. Structure investigations

The results of X-ray diffraction measurements for the MnxSi1-x/ Al2O3(0001) structure (x ≈0.52) are shown in Fig. 6. The diffraction curve contains strong reflection peaks from Al2O3(0006): 2θ = 41.68° for the CuKα1 line, 2θ = 41.78° for the CuKα2 line, and 2θ = 37.5° for the line CuKβ1. In addition to these peaks, this curve contains a broad peak ε fi from the -MnSi(210) lm with B20 structure for the CuKα line at 2θ = 44.43°. The integral characteristic of the film's structural quality is the fi Fig. 7. AFM (a) and MFM (b) images obtained for the Mn0.52Si0.48 sample at room rocking curve width at half maximum (FWHMω). For the lm under fi temperature, i.e. below the Curie temperature of the lm (TC ≈330K). The light regions study, the FWHMω parameter at 2θ = 44.43° is Δω ≈0.4°, whereas the in the MFM image are the attracting magnetic areas. Dark regions are areas in which fi FWHMω value for the single-crystalline lm of the same thickness there is no MFM probe attraction. The arrow shows the interface between crystallites in should be about 250 s of arc. Such a broad peak is a signature of a case when it is poorly appeared in AFM mode (a) while it is obvious in MFM mode (b). pronounced mosaicity and imperfection of the film structure, in Insets in Figs. 7a and b demonstrate the fragments of AFM and MFM images, respectively, at large film surface scanning area 20×20μm2. particular, caused by the lattice constant mismatch between Al2O3 substrate and MnSi film as well as by the Mn excess (for MnSi with B20 structure, the lattice constant a ≈4.56 Å; for Al2O3 a ≈4.76 Å). At increasing Mn content the ε-MnSi(210) peak transforms to a “flat hill” film. Comparison of AFM and MFM images shows that the positions of about 2° wide at x ≈0.63(see inset in Fig. 6). inter-crystallite interfaces correlate on the whole with dark strips in To obtain further insight into peculiarities of the film structure, we MFM images (Fig. 7b). However, the width of these strips (∼0.5μ m) analyze AFM and MFM images of the sample surface (see Fig. 7). The considerably exceeds the width of lines (≤0.1μm) separating blocks AFM and MFM measurements were performed in ambient conditions (Fig. 7a) in the AFM images. Moreover, there exist inter-crystallite for the Mn0.52Si0.48 film demonstrating the above room temperature interfaces which are almost not revealed in AFM images (shown by FM (TC ≈ 330 K, magnetization data are shown in Fig. 3). For receiving arrow in Fig. 7a) in which, however, magnetic heterogeneity (dark the MFM images (Fig. 7b), the two-pass technique (lift-mode) was strips in Fig. 7b) is brightly expressed. used. The height of lift on the second pass was about 30–50 nm. Change of the probe oscillation phase was recorded at the fixed pump 3.3. Possible origin of magnetic anisotropy frequency. The light regions on the MFM images correspond to an increase of the phase arising at a reduction of the probe resonant The itinerant cubic ferromagnet singe-crystal MnSi with B20 frequency which is caused by its attraction to the surface. Therefore, structure has a weak fourth-order cubic magnetic anisotropy. But in light regions on the MFM images display the areas where the probe is case of a thin epitaxial MnSi film deposited on a thick substrate, the attracted to the sample, and dark strips show the areas where such the induced uniaxial magnetic anisotropy can be essential due to the strain attraction becomes weak or is absent. Magnetic images do not depend caused by lattice mismatch between the film and the substrate (see [15] on the sample previous history, i.e. they do not change at preliminary and references therein). magnetization of the sample in this or that direction. The saturation In our case, sufficiently large mismatch (≈4%) between the Al2O3 magnetization of the sample under conditions of measurements substrate and MnxSi1−x film is realized. It is one of the main reasons (T ≈ 290 K) is about 100 emu/cm3, and its coercitivity is a fortiori less for polycrystallinity of the grown film and thus initiates an existence of than 50 Oe (see Fig. 3 and Ref. [6]). In this situation, the local reversal inter-crystallite and crystallite-substrate strain, producing crystal twin magnetization of the sample in the field of MFM probe is possible, planes or inter-crystallite boundaries [16] (below we will use a unified leading to its attraction. term “plane defect”).

According to AFM results, the depth of weakly pronounced inter- XRD measurements clearly show that MnxSi1-x crystallites are block interfaces (thin lines in Fig. 7a) does not exceed 2 nm, while ordered normally to the surface of Al2O3 substrate, i.e. the grown ﬁlms strongly pronounced inter-crystallite interfaces (thick lines) have the are textured (have a mosaic type). According to AFM-MFM images, the form of cavities with the depth of <10 nm at the 70 nm thickness of the characteristic size of crystallite is about 1 µm. Obviously, in the frame

309 A.B. Drovosekov et al. Journal of Magnetism and Magnetic Materials 429 (2017) 305–313 of used methods we are unable to adduce direct experimental proofs of tion of Appendix A is developed for a purely homogeneous case, i.e. the strain inside our films; so, our supposal should be verified in future does not consider distribution of local anisotropy in the plane and on studies. However, it is well known that mechanical strain on the plane the film thickness. Therefore, the constants K1 and K2 found with its defect can induce elastic or plastic deformation (even dislocation) near help have only an efficient character. this defect [16]. Following this paradigm, at least a part of thin lines (one of them is indicated in Fig. 7a by arrow) may be possibly 4. Conclusions associated with the projections of plane defects with elastic strains on the film surface. Some lines (not shown in Fig. 7) have a profound In this work, for thin films of nonstoichiometric Mn Si alloys cavity shape and may be possibly attributed to the plane defects with x 1-x with different manganese content (x ≈ 0.44 − 0.63) the FMR measure- strong non-elastic deformations or dislocations in the film. ments were performed in the wide range of frequencies The MFM data shed light on the magnetic structure of the film ( f =7−60GHz) and magnetic fields (H =0−30kOe). For samples surface, while their interpretation seems to be ambiguous. A single- with x ≈ 0.52 − 0.55, the FMR data confirm the early reported FM domain thin film with strong easy-plane anisotropy does not exhibit order with high Curie temperatures T ∼ 300 K [6,7]. Earlier, we have local reversal magnetization in the field of MFM probe, so the MFM C explained the appearance of such FM order in Mn Si films in frame foreground color should look like homogeneously dark. However, in x 1-x of a non-conventional defect-induced carrier-mediated mechanism [3]. our experiments the MFM signal is locally dark only nearby the lines Further to the fact of FM order itself, studied samples also possibly corresponding to projections of plane defects, as the probe demonstrated in FMR measurements an intricate character of mag- approaches them, while it remains light far from them. A possible netic anisotropy, which can be described in a phenomenological way as reason for such the effect is due to the significant enhancement of a combination of two contributions: the second order easy plane magnetic anisotropy near the plane defects in the film. This enhance- anisotropy component and the fourth order uniaxial anisotropy ment may be provided by an increase of anisotropic (for example, spin- component with easy direction normal to the film plane. In frame of orbit) component of effective exchange coupling between local mag- above mentioned assumption we attribute this magnetic anisotropy to netic moments of Mn-containing nanometer scale defects, due to the existence of a well-pronounced mosaic (polycrystalline) structure of strong crystal potential distortions near the plane defect in the the films. We believe that such a mosaic structure is revealed in nonstoichiometric Mn Si alloy. Following our supposition, the x 1-x presented XRD and AFM measurements, be accompanied by the strain “local” axes of magnetic anisotropy are oriented normally to the plane between crystallites and/or crystallites-substrate. Following our model, defect, i.e. lie in most part in the film plane. The plane defects are these local strain can initiate an enhancement of the spin-orbital randomly distributed in the film, they strongly pin local magnetic anisotropic component of exchange interaction between the local moments of Mn-containing defects and block a local reversal magne- moment on the point magnetic defect and itinerant electron spin in tization in MFM measurements. At the same time, the inhomogeneous the matrix (see Appendix B). This enhancement becomes apparent as a distribution of random “local” in-plane anisotropy axes leads to pinning of local magnetic moments in the MFM images. existence of effective “global” uniaxial anisotropy with symmetry axis We hope that the combination of FMR, XRD, AFM and MFM normal to the film plane. methods showed its efficiency in the study of nonstoichiometric The plane defect driven magnetic anisotropy mechanism does not Mn Si alloys as a new class of high-temperature FM semiconductors qualitatively contradict to the above performed FMR data and their x 1-x with unusual properties. interpretation in frame of the phenomenological description of Appendix A.InAppendix B, we propose a simple quantum mechanical model of randomly distributed plane defects having spin-orbit coupling Acknowledgments with the matrix of a weak itinerant ferromagnet. By means of this model, we support the phenomenological approach of Appendix A.In Authors are grateful to Dr. Vasiliy Glazkov for assistance in carrying particular, the model predicts essential second order easy plane out low-temperature FMR measurements in the region of high fields. anisotropy contribution with K1 >0 and the additional fourth order We express our gratitude to Dr. Alexei Temiryazev for AFM and MFM uniaxial anisotropy contribution with K2 <0. However, the ratio |KK2|/ 1 measurements. We would also like to thank him as well as Dr. Dmitry −3 −4 estimated from the model is |KK21|/ ∼ 10 − 10 ; this is much less than Kholin and Dr. Elkhan Pashaev for fruitful discussions of FMR and the value |KK21|/ ∼0.2−0.4 found from the experiment (see Table 1). XRD results. The possible reason of this disagreement is the used perturbation The work was partly supported by the RFBR (grant Nos. 15-07- approach (see Appendix B) as well as the strain between crystallites 01170, 14-07-91332, 14-07-00688, 13-07-00477, 15-29-01171, 14- and the substrate which is neglected in the proposed model. At the 47-03605, 15-07-04142), NBICS Center of the Kurchatov Institute. same time, it should be kept in mind that phenomenological descrip- The work at HZDR is financially supported by DFG (ZH 225/6-1)

Appendix A. FMR in magnetic ﬁlm with perpendicular anisotropy

Taking into account the perpendicular uniaxial anisotropy, the magnetic energy density of a ferromagnetic ﬁlm is given by the following expression:

2 E =−HM · +2πMzAz + E ( M ), (A.1) where the first term is Zeeman energy in magnetic field, the second term is shape anisotropy of the sample (“demagnetization energy”) and the last term represents general form of uniaxial anisotropy energy with the anisotropy axis oriented along vector z normal to the film plane. Neglecting dissipation, the precession of the magnetic moment is determined by the Landau-Lifshitz equation: ∂M =[γ MH × ], ∂t eff (A.2) where the effective field:

310 A.B. Drovosekov et al. Journal of Magnetism and Magnetic Materials 429 (2017) 305–313

∂E ∂EA Heff =− =−4−HzzπMz . ∂M ∂Mz (A.3) Resonance frequency is defined as eigenfrequency of the system (A.2) after its linearization near equilibrium orientation of the M vector. In sufficiently large fields exceeding the saturation field, the static magnetization is oriented along the magnetic field MH. Taking into account this condition, in case of the field applied in the film plane, the resonance frequency is defined by Eq. (1), where:

⎛ 2 ⎞ ⎜ ∂ EA ⎟ K MπMM=4 + ⎜ 2 ⎟ . ⎝ ∂Mz ⎠ Mz=0 (A.4) When the field is applied perpendicular to the film plane, the resonance frequency is defined by Eq. (3), where: ⎛ ⎞ ⎜ ∂EA ⎟ K⊥MπM=4 + . ⎝ ∂Mz ⎠ MMz= (A.5)

Thus, in the presence of the uniaxial anisotropy the K and K⊥ parameters generally speaking do not coincide. Writing the EAz(M ) function in the form of decomposition:

2 4 6 EA =KθKθKθ1 cos +2 cos +3 cos + …, (A.6) where cosθMM =z / , the formula (A.4) transforms into (2) and the formula (A.5) takes form: 2K 4K 6K K MπM=4 + 12+ + 3 + ⋯ ⊥ M M M (A.7) If the sixth and higher order anisotropy constants are negligible, Eqs. (A.7) and (2) lead to the expression (4).

Appendix B. Microscopic approach

Self-consistent theory of spin ﬂuctuations in the homogeneous itinerant FM [3] is based on the simple model Hamiltonian of interacting fermions:

+ / 0 =()()()+∑ ∫ ΨεΨdααrk rr α ++ +()()(−′)(′)(′)′,∑ ∫ ΨΨUααrrrr Ψ β r Ψdd β rrr αβ, (B.1)

+ where Ψα (r) and Ψα()r are creation and annihilation operators of fermions, ε(k) is fermions spectrum, k =−i ∂/∂r is operator of quasi momentum, (,rr ′) are three-dimensional space vectors, (,αβ) are spin indices, U(−′rr) is effective potential of fermions interaction. Hamiltonian (B.1) is spin- rotation invariant and does not contain relativistic contributions in the fermions spectrum and effective interaction. Let us modify Hamiltonian (B.1) for a case of mosaic film of itinerant FM deposed on the non-magnetic substrate. We suppose that this film is composed of macroscopic grains with characteristic size l in the film plane and nearest neighbored grains are separated one from other by a narrow interfacial region with characteristic thickness d⪡l. Notice that (d,l) significantly exceed the lattice parameter a. The grain boundaries are orthogonal to the film plane throughout all the film thickness and form an arrow of two-dimensional defects inside the film. The crystal potential in the interfacial region significantly differs from the potential inside the grains due to broken chemical bonds, inter grain clustered aggregates, mechanical strains etc. and thus strongly modifies the fermions motion. To model the effect of interfacial boundaries on the fermions behavior in itinerant FM, we treat these boundaries as non-magnetic macroscopic plane defects embedded into the homogeneous matrix and introduce additional term in the Hamiltonian of our system:

+ /1 =∑ ∫ ΨVIδαnn (rD ){[ + ·σρρ ] ( −n )}αβ Ψd β ( rr ) . αβn,, (B.2) fi fi fi Here δx( ) is delta-function, r =(ρ ,z), two-dimensional vector ρn de nes the n-th plane defect position in the lm, axis z is orthogonal to the lm plane, σ is vector composed from the Pauli matrices, Vn and Dn are respectively scalar (Coulomb) and vector (spin-orbit) components of the n-th η plane defect potential. The component Dn may be expressed in the Bychkov-Rashba form as Dnn=[η ek ×], where parameter is proportional to the fi ne structure constant and gradient of interface potential, en is unit vector normal to the n-th defect plane [3]. ff E ect of the bulk magnetic defects with local spins {Sj} on the behavior of fermions in itinerant FM is traditionally considered within the Hamiltonian:

+ / 2 =∑ ∫ ΨJδαjjjαββ (rS ){[σ ] ( rr − )} Ψd ( rr ) . αβj,, (B.3)

Here Jj is corresponding exchange coupling integral, local spins {Sj} are randomly distributed in the three-dimensional fermions matrix with the mean inter-spin distance b, ab⪡⪡(, dl).

From the total Hamiltonian of the system, / =++///012, it is possible to obtain the free energy functional Φ{(),()}mr Sr of itinerant FM with both 2D macroscopic non-magnetic defects and microscopic 3D magnetic defects as a series expansion in terms of the magnetic moment densities of itinerant fermions m(r) and local spins Sr( ). The form of Φ{(),()}mr Sr depends on the studied temperature region and relative ff contributions of di erent terms in the Hamiltonian / . Early in Refs. [3], we analyzed Φ{(),()}mr Sr at the high-temperature region T > TC, where TC is the global Curie temperature of the system, completely neglecting the term / 2, i.e. preserving spin-rotation invariance. Moreover, we revealed in Refs. [3] a drastic enhancement of exchange coupling between local spins of defects due to the effect of itinerant fermions spin fluctuations and derived corresponding expression for TC.

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In this paper, we analyzed Φ{(),()}mr Sr below the global Curie temperature, in the temperature range TSF <

ΔΦ{,mM }={, Φ mM }−{}= Φ0 M

={}+{}+{,},ΦΦδΦ01mm mM 2 4 2 4 ΦAmAmΦBmBm01{}≈mm +2 ,11 {}≈zz +2 , δΦ{,mM }≈ λ mM · . (B.4)

The Φ0{}M term simply shift the free energy scale and does not take interest for us, Φ0{}m and Φ1{}m are respectively isotropic and anisotropic in m contributions of itinerant fermions, δΦ{,mM }is contribution of exchange coupling between itinerant fermions spins and magnetic defects local ﬃ spins. The coe cients in formulas (B.4) can be estimated in the temperature region TSF <

−1 2 BWdlην1 ≈(/)(/),F −3 4 BWdlην2 ≈− ( /)(/F ). (B.6) ff Here W is energy scale of the order of fermions bandwidth, νF is the Fermi velocity, QSF is the cut-o wave vector of itinerant fermions spin fl fl uctuations, ζ(0) ≈νWF / , ζ(T ) is correlation length of itinerant fermions spin uctuations, renormalized by a scattering on the Coulomb component of macroscopic defects potential V. This scattering is not explicitly included in formulas (B.5), since at considered temperatures and relations between the parameters of our model (/)⪡dl 1, [dζ/ (0)]⪡1, (/VW )⪡2 1 it does not lead to new physical effects. Notice, we consider our system as a matrix of weak itinerant ferromagnet with magnetic defects, where sizable spin fluctuations in the matrix lead to a strong enhancement of exchange coupling between local moments of defects. In our approach, the source of defects with local moments in the non-stoichiometric MnxSi1-x alloy with x ≈0.5 is attributed to Si vacancies [7], while spin fluctuations are approximatively treated as “ fi ” paramagnons in the ideal stoichiometric MnSi material. In the mean eld approximation, the Curie temperature TC in the renormalized free fi energy functional Φ0{}m without spin-orbit coupling (see Eqs. (B.4, B.5))isde ned by the condition AT1C()=0, i.e. ζ()→∞TC , and has non-trivial dependence on the electron spectrum parameters and local defect moments concentration. As a result, a significant enhancement of the temperature of onset of long-range ferromagnetic order in the non-stoichiometric MnxSi1-x alloy may be achieved (see Ref. [3]). It is seen from Eqs. (B.5) and (B.6) that coefficients A1, A2, B1, λ are positive in the considered temperature region, while coefficient B2 is negative. Formally this fact is due to an interplay-between different diagrams of the eighth order in Φ{,mM }series expansion in the effective fi perturbation eld [mr()+∑n Dn(−ρρn )], leading to appearance of the fourth order in m anisotropic contributions to Φ1{}m after averaging over macroscopic defects distribution. We do not attribute a profound physical meaning to this result, while it seems noteworthy. fi Varying ΔΦ{,mM }over m, after neglecting for simplicity the Φ1{}m contribution, we get in the mean eld approximation for an equilibrium 22 value m01≈−λA /2M [1−( λAAM 21 / ) ]. Substituting m = m0 in ΔΦ{,mM }and separating isotropic and anisotropic terms, we obtain for the magnetic anisotropy energy EA the expression (A.6) with cosθMM =z / and 2 4 K110∼>0,∼<0Bm K220 Bm . (B.7)

Estimations of relative and a fortiori absolute values of K1 and K2 are uninformative in our model due to their very rough approximate character. Obviously, in our approach contribution of the fourth order terms proportional to K2 must be small compared with contribution of the second order terms proportional to K1 in Eqs. (B.7) due to the used perturbation theory in the ﬁne structure parameter: 2−22−3−4 |KK21|/ ∼( ηνWm / F )0 ∼10 ÷10 ⪡1. (B.8) −1 From the phenomenological model of Appendix A, to satisfy obtained FMR results we have to put |KK21|/ ∼ 10 . Thus, correspondence between Eq. (B.8) and experimental results is not yet good. This disagreement is not surprising since the developed model is based on the perturbation approach to the spin-orbit component of coupling between itinerant electron spin of the matrix and the plane defect moment, i.e. the calculated coeﬃcients K1 and K2 are obtained as the lowest terms in the corresponding series expansion of the coupling energy on the spin-orbit components. For real alloys the used perturbation approach may be incorrect, but unfortunately, in the theory of itinerant ferromagnetism there exist no adequate description to take strong spin-orbit coupling into account.

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